Power amplifier (PA) linearity and efficiency are often two parameters that must be traded off in a wireless system. Fortunately, a Volterra-based adaptive digital-predistortion (DPD) linearizer circuit can enable RF PAs in wireless systems to achieve high efficiency with good linearity. This adaptive digital predistortion solution extends the linear range of power amplifiers and, in combination with crest factor reduction, enables RF PAs to be driven harder and more efficiently while meeting transmit spectral efficiency and modulation accuracy requirements. Part 1 of this two-part series will provide an overview of different digital predistortion techniques and introduce the unique adaptation algorithm that is at the heart of this innovative DPD linearizer.

The new digital predistorter has been incorporated in the model GC5322 integrated transmit solution from Texas Instruments (www.ti.com). The multi-million-gate application- specific signal processor (ASSP) is fabricated with 0.13-micron CMOS technology and incorporates digital upconversion, crest factor reduction, and digital predistortion. The "modulation-agnostic" processor supports signal bandwidths to 30 MHz. It can reduce peak-to-average power ratios (PARs) for thirdgeneration (3G) cellular signals by as much as 6 dB. It provides a 4-dB improvement for orthogonal-frequencydivision- multiplex (OFDM) signals while meeting adjacent-channel-power- ratio (ACPR) and error-vectormagnitude (EVM) specifications. It can correct for up to 11th-order nonlinearities and PA memory effects to 200 ns. It typically provides greater than 20-dB ACPR improvement and a more than four-time increase in power efficiency for a variety of RF PA topologies, resulting in as much as 60-percent reduction in the static power consumption for typical basestations. This flexible Volterra-based predistorter can be optimized for a variety of RF architectures, modulation standards, and signal bandwidths.

Nonconstant-envelope-modulation schemes like those used in 3G and other emerging air interface standards are spectrally more efficient, but have high peak to average signal ratios, necessitating a higher level of PA backoff. This decreases the PA efficiency and increases the cooling and operational costs of the base-stations. Lower efficiency RF PAs typically account for as much as 30 percent of the overall base station system cost and have a considerable environmental footprint. Increasing push toward "green," energy-efficient technologies combined with rising energy costs and increasing spectral efficiency and signal bandwidth requirements of current and evolving wireless standards make power amplifier linearity a crucial design issue in next-generation base stations. A variety of power amplifier linearization techniques such as RF feedforward, RF feedback, and RF/IF predistortion and post-distortion have been proposed and implemented over the years. Of these, adaptive DPD schemes have proven to be the most efficient and cost effective compared to traditional analog/RF linearization techniques. Increasing DSP/ASSP computational capacities make digital pre-distortion an ever more attractive option.

The GC5322 transmit solution combines digital upconversion (DUC), crest factor reduction (CFR), and DPD in a highly integrated ASSP, with real-time adaptation control provided by software residing in a model C67x DSP from Texas Instruments. The transmit device can be optimized for a variety of RF architectures and supports multiple air interface standards including CDMA2000, WCDMA, TD-SCDMA, MC-GSM, WiMAX, and the Long Term Evolution (LTE) cellular standard. The flexible pre-distorter can be used efficiently with a variety of power amplifier topologies such as Class A/B or Doherty amplifiers, and is designed to support communication systems with signal bandwidths as wide as 30 MHz. This two-part article focuses on the hardware implementation of the DPD solution.

Wireless communications systems based on 3G CDMA as well as multicarrier systems using such methods as OFDM often handle signals with high PARs or crest factors. The nonconstant- envelope-modulation techniques such as quadrature amplitude modulation (QAM) employed in such systems have stringent error-vectormagnitude (EVM) requirements. Such requirements call for a PA with highly linear amplitude and phase response. PAs typically have a limited linear range of operation. PA nonlinearities cause intermodulation distortion in the transmitted signal, leading to spectral splatter and reduction in adjacent-channel power ratio (ACPR). A simple solution to this problem is to back off the level of the input signal to the PA so that the resulting output signal lies completely within the amplifier's linear operating region. Unfortunately, PA power efficiency decreases considerably at lower input power levels, making this a less-thanoptimal solution. Moreover, even advanced, efficient amplifier topologies such as Doherty PAs suffer considerable nonlinearities even at backed-off power levels, resulting in poor EVM and ACPR performance.

The efficiencies of traditional Class AB power amplifiers in use today range from about 5 to 10 percent when operated under back-off conditions. But with crest-factor reduction and adaptive DPD techniques, the efficiency can be improved by a factor of 3 to 5. Newer PA topologies, such as Doherty amplifiers, or even Class AB amplifiers with dynamic envelop tracking in combination with DPD, and newer device technologies, such as gallium nitride (GaN) or gallium arsenide (GaAs) power transistors, can be used to achieve efficiencies approaching 50 percent.

Current DPD implementations mostly use memory-less linearization techniques in which an instantaneous nonlinearity (the predistortion) is used to compensate for the instantaneous nonlinear behavior of the PA. Memory-less power amplifiers can be characterized by their amplitude and phase transfer characteristics, commonly referred to as AM-to-AM (or gain compression) and AM-to-PM characteristics. A generalized lookup table (LUT) can be used for the predistorter gain/phase correction for such a memory-less power amplifier. *Figure 1* shows the gain compression and AM-PM characteristics for a typical Doherty PA. Because the gain and phase characteristics of a PA change with temperature, voltage, and component aging, adaptive control of the lookup tables is required for truly efficient and effective linearization.

For communication systems where the PA must support higher RF modulation bandwidths, the memory-less model proves to be inadequate since it is only amplitude dependent, not frequency dependent. PAs that must support large signal bandwidths exhibit significant memory effects due to the long time constants of components in the DC biasing networks and rapid thermal effects of the active devices. This causes the PA's characteristics to vary as a function of earlier input levels, and necessitates the use of a predistortion architecture that can alleviate these memory effects.

Any efficient linearization scheme requires a highly accurate model for the predistorter, as well as for the PA if it uses a direct learning adaptation architecture. A variety of techniques have been proposed in the literature for modeling nonlinear systems with memory, with none providing a universal solution. As a result, model selection is challenging and dependent on the application. An efficient PA model must represent different types of non-linearities and memory effects in PAs with reasonable accuracy.

One of the more general models for time-invariant nonlinear systems with memory is the Volterra series. It consists of a sum of multidimensional convolutions, which in discrete time causal form can be written in the form of Eq. 1 with the conditions detailed in Eq. A, where the multidimensional matrices h1, h2, hn are the nth-order Volterra coefficients that model the nonlinearities, and Mn is the finite-length memory of the nonlinearity. With the long memory depths (to 1 microsecond) and nonlinearity orders (to 11th order) to be considered for RF PAs, the above model becomes computationally intractable. Simplification schemes must be employed to yield a practical predistorter product. These simplifications can be placed into two basic approaches: algorithmic approaches and model-reduction approaches. For the first set of approaches, the generic Volterra model in Eq. 1 has a number of attractive arithmetic properties that can be exploited to come up with efficient implementations. For the modelreduction approaches, although a totally generic Volterra (or some other generic model) is desired, it is known that RF power amplifier models typically have a lot of Volterra terms that are insignificant for practical implementation. The terms may be dropped without measurable degradation of linearization performance.

A variety of different simplified pre-distortion systems, all using variations of the generalized model in Eq. 1, have been proposed in the current literature. A few of these systems are listed here:

* 1. Truncated Volterra^{3,5}* Direct-form, parallel-cascade, Vvector algebra based and a few other realizations of truncated Volterra systems have been proposed in the literature. These algorithmic reduction approaches are efficient at linearization, but are computationally complex and often intractable due to the large number of parameters to be estimated, making them unattractive for real-time implementations.

* 2. Wiener systems^{6,7}* A significant simplification of the Volterra model, the Wiener model consists of a linear filter followed by a memory-less nonlinearity. An LUT can be used to model the nonlinearity, and an FIR filter to model the linear filter. The effectiveness of Werner systems in modeling most RF power amplifiers is limited. The estimation of the model parameters is reasonably complicated, making it unattractive for real-time adaptation.

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* 3. Hammerstein systems^{6,7}* Again, a reduction of the Volterra model, the Hammerstein model consists of a memory-less nonlinearity followed by a linear filter. It is a simple memory nonlinear model, and it is easier to compute its model parameters than for a Wiener model. This model is also of limited effectiveness for modeling all different types of RF power amplifiers.

* 4. Wiener-Hammerstein^{6,7}* Cascading a linear filter, a memoryless nonlinearity and another linear filter form a Weiner-Hammerstein model. This model is more general than a Weiner or a Hammerstein model, and covers a lot more terms from the Volterra series enabling better modeling of the non-linearity.

* 5. Memory polynomial^{2,4}* Constraining the Volterra series in (1) so that everything except the diagonal terms in the kernels are zero, i.e. hn(i1,i2,i3) != 0 only when i1=i2=i3, creates a memory polynomial model as shown in Eq. B below, where M is the memory length and K is the nonlinearity order.

This model (and its variations) has been shown to be effective at linearizing wideband power amplifiers, and have reasonable hardware and software computational requirements.

Various combinations of the above models have also been suggested in the literature, each with its own pros and cons. A commercially viable predistorter needs to be adept at tackling a wide variety of non-linear behaviors, and might require different modeling schemes for different applications. For most of these models, the predistorter coefficients are adapted with an indirect learning architecture using least squares identification.

In the GC5322 predistorter implementation, a combination of algorithmic and model reduction approaches are used for a tractable realization. The number of terms in Eq. 1 can be significantly reduced by eliminating redundancies associated with various index permutations. The Volterra coefficients can be assumed to be symmetric without any loss of generality. Furthermore, the real input signal to the power amplifier, x(n), can be expressed in terms of its complex baseband representation, x(n) = Re{e^{jx}_{0}^{n}X(n)}, where Ω_{0} = 2 π f_{0} and f_{0} is the center frequency of the frequency band of interest.

Since for band-limited systems only frequency components close to the carrier frequency f_{0}, are of interest, writing the Volterra series in terms of complex baseband signals will help in significantly reducing the number of terms to consider and help guide the choice of model architecture. For example, the even-order intermodulation terms will lie far away from the frequency band of interest, making it possible to further drop one-half of the terms in Eq. 1. The model is rotationally invariant, which allows for further simplifications. This means that a phase shift on the input of the PA produces exactly the same phase shift on the output. The implication is that Eq. 1 can then be reduced to terms involving products of the signal and powers of its magnitude squared. Moreover, it is known that the PA is causal, and it is assumed that the linear portion of the PA is minimum phase (or sufficiently so). This further restricts the Volterra terms.

In most PAs, signal processing is performed in stages. By exploiting this feature, the model can be simplified (in the number of terms required for a given application) into cascaded sections with each matched to the needs of compensating the distortion in each PA stage.

The DPD implemented in the GC5322 is split into these three major blocks: the linear equalizer, the nonlinear DPD, and the feedback nonlinear compensator and smart capture buffers. By restricting the Volterra series in Eq. 1 to only the linear terms with memory M1, the model for the linear equalizer block (Eq. 2) results:

*Y1(n) = Σi=0:M1 h1(i).x(n-i)* (2)

A (M1-1)-taps-long complex transmit equalizer can account for linear distortion in the RF transmit path and the PA. It can be considered as the linear time-invariant half of a Hammerstein model. This equalizer primarily compensates for filtering in series with the PA, such as matching networks, duplexers, and IF filtering. The equalizer implemented in the GC5322 provides a correction time span from 100 to 200 ns depending on the clock rates chosen. This places a maximum amplitude and group delay constraint on the analog design. A 2-ns peak-to-peak group delay and a 1-dB peak-to-peak amplitude ripple specification for the analog portion of the transmitter design was deemed to be a reasonable compromise between analog and digital complexities. The hardware implementation of Eq. 2 provides a complex FIR filter on both the real and imaginary datastreams. This allows for independent equalization of the real and imaginary signal paths, and can compensate for I/Q gain/phase/delay mismatch.

The second block of the transmit ASSP is the nonlinear DPD. It is required since nonlinear memory effects in PAs can range from a couple of nanoseconds to as much as one microsecond depending on the PA design and signal bandwidth. Combined with high orders of nonlinearity in PAs for wireless systems (from 5th order for Class AB amplifiers to as high as 11th order for Doherty PAs), the selection of a suitable nonlinear predistortion architecture can be truly challenging.

Simplifying the Volterra series in Eq. 1 by restricting it to only the nonlinear diagonal terms with memory M2, and dropping even terms as mentioned above, yields the model for the nonlinear predistorter block shown in Eq. 3.

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This predistorter block can account for the bulk of the PA nonlinearities. If memory is ignored for this block, it can be considered as the memory-less nonlinearity portion of a Hammerstein model. With memory included, it can be used as a memory-polynomial based predistorter. Rearranging terms results in the relationship of Eq. C.

This rearrangement of terms has reduced the equation to a finiteimpulse- response (FIR) form, and makes it possible to implement the polynomial in |x(n-i)|^{2} in a hardwareefficient LUT form. The order of the polynomial is limited by the modeling accuracy tolerance of the adaptation algorithm.

For some types of RF PAs, additional memory effects are dependent on the signal envelope history. These memory effects, for example, could be due to a number of different factors, such as thermal and power supply transients that act nearly as a multiplicative gain that is a function of the power history. The terms from the Volterra series in Eq. 1 that involve cross products of the signals to be amplified and their complex signal envelope can be used to form a relationship useful in exploring the memory effects of RF PAs and how filters can be used to achieve improved linearity.

*Editor's Note: This ends Part 1 of this two-part article, which will conclude with the August issue of Microwaves & RF, demonstrating the application of DPD for linearizing an actual RF PA for 3GPP systems.*

REFERENCES

1. R. Sperlich, Y. Park, G. Copeland, and S. Kenney: "Power Amplifier Linearization with Digital Pre- Distortion and Crest Factor Reduction," Proceedings of the 2004 IEEE Microwave Theory & Techniques Symposium, pp. 669-672.

2. J. Kim and K. Konstantinou, "Digital Predistortion of Wideband Signals Based on Power Amplifier Model with Memory," *Electronic Letters*, vol. 37, No. 23, 2001, pp. 1417-1418.

3. T. Panicker and V. Mathews, "Parallel-Cascade Realizations and Approximations of Truncated Volterra Systems," *IEEE Transactions on Signal Processing*, vol. 46, No. 10, 1998, pp. 2829-2832.

4. D. Morgan. Z. Ma, J. Kim, M. Zierdt, and J. Pastalan, "A generalized memory polynomial model for digital predistortion of RF power amplifiers," IEEE Transactions on Signal Processing, vol. 54, No. 10, 2006, pp. 3852-3860.

5. A. Jhu and T. Brazil, "An adaptive Volterra predistorter for linearization of RF high power amplifiers," Proceedings of the 2002 IEEE Microwave Theory & Techniques Symposium, pp. 461-464.

6. P. Gilabert, G. Montoro, and E. Bertran, "On the Wiener and Hammerstein Models for Power Amplifier Predistortion," Proceedings of the 2005 IEEE APMC.

7. A. Shah and B. Jalali, "Adaptive equalization for broadband predistortion linearization of optical transmitters," *IEEE Proceedings on Optoelectronics*, vol. 152, No. 1, 2005, pp. 16-32.